Plünnecke inequality ($|X+mB|\le K^m|X|$)

For nonempty finite subsets $A,B$ of an abelian group, $|A+B|\le K|A|$ implies that some nonempty $X\subseteq A$ satisfies this bound simultaneously for every integer $m\ge 0$, with $0B=\{0\}$. The Petridis minimal-growth lemma proves this by choosing $X$ to minimize $|X+B|/|X|$. The Ruzsa triangle inequality then gives the Plünnecke-Ruzsa inequality $|kB-\ell B|\le K^{k+\ell }|A|$.